Geometric Bites

Continuity and Differentiability (Clear, Compact Guide) We study functions of a single real parameter u . A scalar function is written as φ(u) , and a vector function as R(u) = (x(u), y(u), z(u)) . All the ideas below are based on ordinary one-variable limits. 1) What it means for a function to be continuous at u (scalar case) A function φ is said to be continuous at u if its value changes smoothly as u changes slightly. Formally, for every ε > 0 , there exists a δ > 0 such that |φ(u + Δu) − φ(u)| < ε whenever |Δu| < δ . 2) Continuity of a vector function Let R(u) = (x(u), y(u), z(u)) . The function R is continuous at u if each component x(u) , y(u) , and z(u) is continuous at that same point. Equivalently, using any fixed norm |·| on ℝ³ : for every ε > 0 , there exists a δ > 0 such that |R(u + Δu) − R(u)| < ε whenever |Δu| < δ . 3) Differentiability (first order) A scalar or vector function is differentiable at u if the limit (F(u + Δ...

How to Derive the Derivative of a Vector Function 🧮 Let’s start with a vector function of a single variable: R(u) = x(u)î + y(u)ĵ + z(u)k̂ Here, x , y , and z are differentiable scalar functions of a real parameter u . This means that as u changes, the point R(u) moves through space — tracing a smooth curve. ✨ The Goal We want to find dR/du — the rate at which the vector R(u) changes with respect to u . Proof By definition of the derivative: dR/du = lim Δu→0 [R(u+Δu) − R(u)] / Δu = lim Δu→0 [x(u+Δu)î + y(u+Δu)ĵ + z(u+Δu)k̂ − (x(u)î + y(u)ĵ + z(u)k̂)] / Δu = lim Δu→0 [(x(u+Δu)−x(u))/Δu]î + [(y(u+Δu)−y(u))/Δu]ĵ + [(z(u+Δu)−z(u))/Δu]k̂ = (dx/du)î + (dy/du)ĵ + (dz/du)k̂ Interpretation The derivative dR/du is itself a vector — one that points in the direction of motion of R(u) and whose magnitude gives the speed of change. Each component ( x , y , z ) behaves just like an ordinary function, so we can differentiate them individually and recombine t...